Balance Sheets: Equity vs. Retained Earnings net, question, balance, correct

bigredhockey55

I’m a little confused about how Equity and Retained Earnings work.  I just answered a QBank question that asked me to calculate ROE based on the balance sheet and income statement, where net income was 4127, common stock was 4000 and retained earnings was 4354.  I would have thought that one just did (net income / common stock), but the correct calc is (net income / (stock + retained earnings)).  
Can somone clarify Equity to me.  I think what confuses me the most is why Equity is listed on a balance sheet in the first place.  Isn’t just a calc of Assets - Liabilities?  It doesn’t seem to exist concretely without these.  I seems circular and redundant to need to show it on a balance sheet.
The way I’ve explained equity to myself is to think of it like my house and my mortgage.  I have a $500K house, my mortgage is $400K, therefore I have $100K equity.  I get that.  But then how does the retained earning fit into that?  Does the analogy flow that if I had rented a room (say for $4K a year) and had made $20K from that and not spent it, that would be retained  earnings, so therefore the $100K of equity is $80K stock and $20K retained earnings shown on my balance sheet?   So to calculate my ROE, I would do $4K / ($100K).  
If the above is correct, what is the point of stripping out the retained earning from the stock on the balance sheet?  Is it to display how much $ have come from initial investments vs. how many $ have come from earnings?